Use cone height for volume and slant height for curved area
Use (1/3)πr²h for a right circular cone · Use πrl for its curved area · Separate radius, vertical height and slant height
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Circle areas
- Pyramid volume and right triangles
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use (1/3)πr²h for a right circular cone
- Use πrl for its curved area
- Separate radius, vertical height and slant height
Two heights, two jobs
A right circular cone has volume (1/3)πr²h, with perpendicular height h. Its curved surface area is πrl, with slant height l. Add the circular base πr² for a closed total surface area. The two heights are related by l²=h²+r² for this right cone, but they cannot be interchanged.
Read the axial section
The diagram shows an axial cross section with radius 3 cm, perpendicular height 4 cm and slant height 5 cm. Its full base diameter is 6 cm. Cone volume is 12π cm³, curved area 15π cm² and closed total area 24π cm². The pictured triangle is a section through the solid, not its complete curved-surface net.
Sources for this lesson
Pause and explain
Which cone measurement is used in πrl?
Worked example
For r=3 cm, h=4 cm and l=5 cm, find cone volume and total closed area.
Show the worked solution
- Volume=(1/3)π×9×4=12π cm³.
- Curved area=π×3×5=15π cm²; add base 9π cm².
Answer 12π cm³; 24π cm².
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A right cone has r=3 cm, h=4 cm and l=5 cm. Find volume and curved area.
Give me a hint
Use different heights for the two quantities.
Compare my reasoning
- Volume=(1/3)π×9×4=12π cm³.
- Curved area=π×3×5=15π cm².
12π cm³; 15π cm².
Look for these in your work
- Choose the correct height
- Keep volume and area units distinct
A right cone has radius 5 cm, perpendicular height 12 cm and slant height 13 cm. Find its closed total area and volume.
Give me a hint
Add a circular base to the curved area.
Compare my reasoning
- Area=π×5×13+π×25=90π cm².
- Volume=(1/3)π×25×12=100π cm³.
90π cm²; 100π cm³.
Look for these in your work
- Include the base when closed
- Use the perpendicular volume height
A conical roof has base diameter 6 m, perpendicular height 4 m and slant height 5 m. Find the exposed roof area when the circular base is not covered.
Give me a hint
Use radius 3 m and curved area only.
Compare my reasoning
- Curved area=π×3×5=15π m².
- The circular base is absent from the roofing surface, so do not add 9π m².
15π m².
Look for these in your work
- Use radius rather than diameter
- Count only the requested surface
Common mistakes
- Volume uses perpendicular height, not slant height.
- The section triangle is not the curved surface area.
What can you explain now?
Which cone measurement is used in πrl?
Compare with the explanation
The slant height l.
It measures along the curved surface from rim to apex.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Which cone measurement is used in πrl?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.